Only if there is a reorg that is sufficiently long1. In order for that to happen, you need more hash power working on the other side of a fork than the hash power that is working on the current best chain.

2. Can we remove the testnet block explorer given that one for mainnet is available?¶

Testnet will continue to be a testing ground for new features, so it makes sense to keep the testnet block explorer2 up.

It just means a block that you do not have the parent for was received3. It pretty much always happens when restarting dcrd due to the way the sync works. You are still syncing up to the latest block, but the remote peer, that is ahead of you, sent a notification that a new block showed up, one for which you do not have the parent yet.

Rather than ignoring it, once you are sufficiently close to being synced, it essentially saves them for later so once you do have the parent blocks those newer blocks automatically get connected.

4. Should I download the blockchain again with every new software release?¶

You should almost never have to download the chain again4 on an upgrade unless it is just a super massive change where it makes sense to start over. Typically though, it should migrate things so it just works.